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There is so much wrong here, and what's more, since you are asking complete strangers on the internet, you already know this is wrong. Figure something else out.
From a legal strandpoint, they don't need a written contract to enforce their position, which seems to be at the crux of your argument.
You entered into the contract in what's known as a course of dealing. Alternatively, an implied contract was formed when you accepted the check. Finally, this entire line of thinking is honestly despicable. Not judging here, but you set the deal up, Navy came through for you, and the best thing to counter your crappy selection of vehicle is to screw them over on some bogus technicality?? I think you can be bigger than that. You are your actions; surely you don't want this as a defining moment in your life, do you?
Perhaps watch the storm scope as hurricane season is soon approaching; maybe you can park the car in the hurricane's path? Or perhaps it failed to start down near the storm surge... :-) <--just kidding.
Since you accepted the loan, that alone is enough to enforce the agreement (though it would be a different story if you didnt accept it). You can have them repo the car, but you will still be liable for the difference after the car is sold (or if its not sold, the whole amount). Its best to be the bigger person and not try to use the excuse of not signing loan documents as a way to escape your responsibilities. That wont hold up in court if they were to sue you, and in fact, since you didnt sign the loan documents and try to bring that up to them, they could demand the whole balance to be due right then and there. If you are having hardship talk to the lenders and see if there is any options for you. If you dont want the car, you can give it to them but that wont resolve you of your liabilities and can affect your credit.
I would not rely upon any legal opinions solicited on a forum.
You have a contract issue for which you should obtain the opinion of an attorney.
What do you have in writing from them?
An Online Application?
The Loan Docs DO need to be signed for them to be able to get a judgement... get a lawyer....